Chapter 5: Strange Attractors

This chapter presented numerous concepts that I found interesting. It is very detailed and long, so I will split the topics into two sections.   

The first topic presented dealt with turbulence, and the way it presents interest that is usually one-sided. Many people are in favour of its disappearance, while few others give supporting evidence for its importance. In some cases turbulence is desirable- for example, inside a jet engine, where efficient burning depends on rapid mixing. However, in most cases turbulence leads to

disaster. For example, turbulent airflow above an airplane’s wings destroys its lift. So, what really is turbulence? It is a mess of disorder that drains energy and creates drag. The process by which flow changes to turbulence is quite interesting. When something shakes a fluid, the fluid starts to move vigorously, as energy drains out of it.  As the liquid shakes, energy is added at low frequency, or large wavelengths, and the large wavelengths decompose into small ones. This is when eddies form, as they dissipate the fluid’s energy. An eddy is the swirling of a fluid and the reverse current created. Knowing all this, there is the question of what happens just when turbulence begins. The theory to understanding this question came from Lev D. Landau, a Russian scientist. His theory on fluid dynamics explained how when more energy comes into a system, new frequencies begin one at a time, such as how a violin responds to harder bowing by vibrating with a second, dissonant tone, and goes on until the sound gets louder. In Landau’s view, wild motion, such as oscillation, simply accumulate one on top of the other, creating rhythms with overlapping speeds.

The rest of this chapter goes on to explain a different idea- phase space and the strange attractor. The strange attractor is a complex attractorwith chaotic motion, which lives in phase space. Phase space 

is a space in which all possible states of a system are represented. It gives way to obtaining essential information from a system of moving parts and making a road map to all its possibilities, and predicting future behaviour. One point in phase space contains all the information about the state of a dynamic system at any instant. In phase space, when looking at a pendulum, with friction as a factor, a central point, where velocity is zero, “attracts” the orbits, as friction takes away some of the energy from the system. This is an example of an attractor, which exhibits predictable behaviour. A strange attractor in phase space is a very interesting concept. The situation is presented when a bounded chaotic system has a long-term pattern that is not a simple periodic orbit. If this situation is plotted on a graph over an extended period, patterns that were not obvious in the short term will be apparent as the system attempts to reach equilibrium. A strange attractor represents a path where a system changes from system to system without settling down. Like snow flakes, they come in great variety and no two are alike.

Chapter 4: Geometry of Nature

Benoit Mandelbrot examined data drawn from rivers. He obtained data from Egyptians on the height of the Nile River for millennia. He found that the Nile had unusually great variation, as it flooded in some years, and subsidized in others. Mandelbrot classified the variations he observed using the Noah and Joseph Effects. The Noah effect meant discontinuity: when a quantity changes, it was change very fast. The Joseph Effect meant persistence: change over time was constant. These Effects explained how trends in nature are real and can vanish very quickly. Mandelbrot also studied geometry, which when applied to the real world is not perfect. It mirrors a rough universe, that doesn’t perfectly mirror the accepted geometric shapes. For example, mountains are not cones and lightning does not travel in a straight line.  

Since Euclid measure

ments – lengths, depth, thickness- failed to capture irregular shapes, Mandelbrot turned to idea of dimensions, then fractional dimensions. A fractional dimension is a way of measuring quantities that do not otherwise have a clear definition. For example, measuring the degree or definition of roughness or brokenness, or irregularity in an object. Fractal dimensions, as they were soon called, found applications on a series of problems that connected to surfaces that were in contact with each other. Several examples include the contact between tire treads and concrete, the contact between joints, and contact in electricity. Then there are the equations of fluid flow, which are dimensionless, meaning they do not require a set scale. Blood vessels, from aorta to capillaries, form a continuum as they branch and divide until they have to move in a single file. With the nature of its structure, the circulatory system performs dimension magic, as they it manages to squeeze a large surface area into a small volume. Yet, the blood only takes up five percent of the body. This is one thing I found very fascinating. How did the body evolve to form such a perfectly complex system?

Chapter 3: Life's ups and downs

This chapter looks at fluctuations in nature, and the work of population biologist as they started incorporating chaos as a factor in changing wildlife populations.  It also looks how hard it is to come up with an equation complex enough to represent real phenomena.

First when analyzing a population a few factors pollution biology looks at are connected to the history of life, looking at how predators and prey interact, and how the spread of a disease is affected by population density. 

 Another way of analyzing a population over time is through modeling the data with an appropriate function. The ideal function to model the growth of a population over time would give way to many factors and allowthe population to settle into its long-term behaviour. One equation that was derived to represent population growth is X_{next=rx(1-x). The parameter r represents the rate of growth, (1-x) keeps the growth within reasonable bounds, since rises, (1-x) falls. Population is then expressed as a fraction between one and zerp. Zero representing extinction, and one representing the largest possible population in the particular environment. After calculating for high parameters, biologists began to notice a higher fluctuation of populations. Many scientists at this time just thought this chaotic behavior was because of malfunctioning calculators, as they noticed that populations tend to rise sharply and fall dramatically before reaching an equilibrium. But, this fluctuation of results could have been attributed to the oversimplified equations that were not capable to take into account all the factors of nature that influence a system. It would require nonlinear equations that would have the necessary variables, but would be extremely difficult to solve. In secondary school, students are only taught how to solve linear systems that are solvable. Non-linear systems with real chaos are rarely understood, as people are constantly trying to make sense of the world and not account for all the disorder amongst them. Robert May, a biologist, further reasoned for the necessities of students to learn about chaos. May believed proclaimed, “ The mathematical intuition so developed ill equips the students to confront the bizarre behaviour exhibited by the simplest of discrete nonlinear systems.” One example May looked at that explained the need for chaos to be understood by scientists in all fields was the debate of population change.  Some ecologists argued that populations are regulated and steady, while others argued that populations fluctuate erratically. May believed that the answer to this argument could be explained though chaos theory. The message that simple models could produce what looked like random behaviour, but was in fact behaviour with fine structure. It was like seeing disorder in an ordered system.}

Chapter 2: Revolution

The next chapter discusses the continuous path of chaos, and the difficulties of communicating new ideas to skeptical scientists. As more people were involved in exploring the new science, scientists were willing to test and apply the theory to other applications in the world.

One tool that scientists used to apply the new science was a pendulum, which can be applied to many real life situations. Galileo observed a church lampswaying back and fourth, and Christian Huygens used the pendulum as a way of timekeeping. Foucault, the Pantheon of Paris, used a pendulum to demonstrate the earth’s rotation. Also,

 every clock, until the era of 

the vibrating quartz, relied on a pendulum. Basic electronic circuits are described by the same equations that describe a swinging pendul

um. Galileo saw regularity in the pendulum, and contended that a pendulum not only keeps precise time, but keeps the same time no matter h

ow wide the angle of its swing. In other words, a wide-swinging pendulum has farther to travel, but happens to travel that much faster.  Galileo’s theory was so convincing, that it is still taught in some physics books, but it is not correct, as was proved later on. The consistency Galileo saw was only an approximation. The changing angle creates equations that are slightly nonlinear. At low amplitudes, the error is almost nonexistent, but as the amplitude increases, the error margin increases. In his experiments, Galileo disregarded friction and air resistance, which play a role in the motion of a pendulum. For example, when looking at a playground swing, the swing accelerates on its way down, and decelerates on its way up, while losing some speed to friction. As the swing gets a push from behind, it accelerates and 

eventually settles back into steady motion. There is friction acting on the swing pulling it down. Also, it can be observed that on a swing as the angle decreases, the swing slows down and goes back and fourth less times that when is at a higher angle.

 

The second part of chapter one looks at the mystery of the Great Red Spot of Jupiter, which is a vast, chaotic, swirlig oval. Astronomers noticed a blemish on the planet, and there were multi

ple theories on what the coloration was. Here are a few examples:

The Lava Flow theory: Scientists imagined a huge oval lake of molten lava flowing out of a volcano.

The New Moon Theory: A German Scientist suggested that the spot was a new moon emerging from the planet’s surface.

The Egg Theory: Some scientists thought the spot was a solid body floating in the atmosphere just like an egg floats in water

When the Voyager orbited in 1978, astronomers saw the spot as a hurricane-like system. However, it possessed some characteristics that were unlike a hurricane.

The final conclusion was that The Great Red Spot was a great anti-cyclonic storm. As hot gases on Jupiter’s atmosphere swirl, and cooler gas falls through the atmosphere, the Corolis effect causes the region to start swirling. These small swirling storms move around and eventually come together, combining their energies and forming the Great Red Spot. The Great Red Spot persists over time because there is no solid ground to create frictions, and slow down the storms. The chaotic flow created takes up much energy, and although the spot itself is a self-organizing system, created by the same factors that create he unpredictable twists, it is stable chaos. It is very interesting how the storm has been going on for over 40 years. I wonder when it will finally die 

down. And will it ever die down?

Chapter 1: The butterfly Effect

The discovery of the new science, Chaos, brought on much skepticism from established scientists. This new notion, Chaos, can be defined as the existence of unpredictable or random behaviour. The study of chaos was the beginning of analyzing disorder in the atmosphere, in turbulent seas, in fluctuations of wildlife populations, and in oscillations of the heart and the brain. In the first chapter of Chaos, by James Gleick, the reader learns about Lorenz,a meteorologist who built a weather simulation program on his computer. This program mirrored real world weather trends, until one day Lorenz made a small rounding error while entering his data, and skewed all his results. From his small miscalculation, Lorenz discovered that the weather cannot be predicted long term because small disturbances, such as rounding to a wrong decimal or the disturbance of the air by a butterfly, was enough to change weather patterns.  

From this, he went on with his extensive research and discovered the phenomenon known as “sensitive dependence on initial condition”, which is referred to as the butterfly effect. The term butterfly effect refers to the idea that a butterfly flapping its wings can create small changes in the atmosphere, which may alter the path of a tornado. The butterfly flapping its wings represents a small change in the initial state of a system, which causes a chain of events leading to larger chaotic events. While the butterfly itself doesn’t case the tornado, the flapping of its wings is part of the initial condition of the system 

I find this very interesting because it shows how small actions, such as a butterfly flapping its wings, or a person making a decision to tie his shoe, can have larger results that influence the lives of many people.  How is it possible that such a small animal and such small movements in the air can drastically change events on ourplanet?

 

Figure 1: Lorenz Attractor: Example of butterfly effect representation